No. We can only see as far as photons could possibly have traveled since the universe originated. This distance is commonly referred to as the particle horizon. On a more practical note, we can only see back as far as photons were able to freely travel through the universe. This is commonly referred to as the surface of last scattering which came into being when the universe was about 380,000 years old. We are pretty confident the universe existed prior to its 380,000 birthday, but, we cannot detect photons emitted prior to that time because the universe then was filled with a hot plasma that is opaque to photons . We could, however, in theory detect neutrinos and gravitational waves emitted when the universe was very much younger. Neutrinos were emitted when the universe was only a couple minutes old and gravitational waves should have been emitted about the same time as when the big bang occurred.

Is there more to our universe than what we can observe? If so, does that mean that photons from the CMB are traveling towards us from beyond our cosmological horizon?

To expand a little bit upon Chronos point, over time we'll be able to see CMB photons that were emitted from a little bit further away. But not very much further. Due to the way the expansion is likely to progress in the future, we'll only ever be able to see a finite distance.

No. We can only see as far as photons could possibly have traveled since the universe originated. This distance is commonly referred to as the particle horizon. On a more practical note, we can only see back as far as photons were able to freely travel through the universe. This is commonly referred to as the surface of last scattering which came into being when the universe was about 380,000 years old. We are pretty confident the universe existed prior to its 380,000 birthday, but, we cannot detect photons emitted prior to that time because the universe then was filled with a hot plasma that is opaque to photons . We could, however, in theory detect neutrinos and gravitational waves emitted when the universe was very much younger. Neutrinos were emitted when the universe was only a couple minutes old and gravitational waves should have been emitted about the same time as when the big bang occurred.

Ok that makes more sense to me. Then as a follow up question, are the photons that we continue to see from the CMB coming from further and further away? If so, does that mean that our cosmological horizon is growing?

Yes, CMB photons we observe in the future will originate from more distant regions of the universe. This, however, has little to do with the cosmic event horizon. Curiously enough, while the particle horizon never stops increasing, the cosmic event horizon actually shrinks as the universe ages! This is because the particle horizon applies to photons emitted at t = 0 and the cosmic event horizon applies to photons emitted at t = NOW. About 5 billion years ago, expansion of the universe began to accelerate. This means photons emitted NOW in regions that are already receding superluminally will never be able to reach a region that is receding subluminally, and hence us. CMB photons, however, were emitted long before expansion began accelerating so they will eventually reach us.

Yes, CMB photons we observe in the future will originate from more distant regions of the universe. This, however, has little to do with the cosmic event horizon. Curiously enough, while the particle horizon never stops increasing, the cosmic event horizon actually shrinks as the universe ages! This is because the particle horizon applies to photons emitted at t = 0 and the cosmic event horizon applies to photons emitted at t = NOW. About 5 billion years ago, expansion of the universe began to accelerate. This means photons emitted NOW in regions that are already receding superluminally will never be able to reach a region that is receding subluminally, and hence us. CMB photons, however, were emitted long before expansion began accelerating so they will eventually reach us.